Performance of Inflation Targeting Based on Constant Interest Rate Projections

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* _{An earlier version was presented in the workshop on “Expectations, Learning and Monetary Policy”} organized by the Deutsche Bundesbank and Center for Financial Studies (CFS). We thank Paul Mizen for helpful discussions, which led to this paper. We also thank Giuseppe Ferrero, Charles Goodhart, Athanasios Orphanides, Lars Svensson, Anders Vredin and the referees for useful comments. This work in part done when the first author was associated with Research Unit for Economic Structures and Growth, University of Helsinki. Support from the Academy of Finland, Bank of Finland, Yrjö Jahnsson Foundation and Nokia Group is gratefully acknowledged.

CDMC04/06

Performance of Inflation Targeting Based on

Constant Interest Rate Projections

Seppo Honkapohja

University of Cambridge

Royal Holloway College,

Kaushik Mitra

University of London

J

2004

A

Monetary policy is sometimes formulated in terms of a target level of inflation, a fixed time horizon and a constant interest rate that is anticipated to achieve the target at the specified horizon. These requirements lead to constant interest rate (CIR) instrument rules. Using the standard New Keynesian model, it is shown that some forms of CIR policy lead to both indeterminacy of equilibria and instability under adaptive learning. However, some other forms of CIR policy perform better. We also examine the properties of the different policy rules in the presence of inertial demand and price behaviour.

JEL Classification: E52; E61; E32.

Keywords: Indeterminacy, instability under learning, inflation targeting, inertia in demand, inflation inertia.

1

Introduction

Inflation targeting has become a fairly common objective of monetary policy in the past ten to fifteen years; see e.g. (Svensson 2003a). This general objective can be implemented in a number of different ways. One possibility is to formulate an explicit objective function, i.e. a “general targeting rule” using the terminology suggested by Lars Svensson. A different approach has been the use of a particular target level for the inflation rate. This target is usually specified at some given horizon for the future and we may speak of inflation forecast targeting in this case. Formally, the inflation target is set for some fixed forecast horizon h and policy tries to achieve that target:

Etπt+h = ¯π. (1)

HereEtπt+h is the forecast of the inflation for periodt+hand in the analysis

it will be taken to be the rational expectations (RE) forecast. In other words, the central bank acts as if private agents have RE, which are computed under knowledge of the structural model of the economy.

It can be noted that if the horizon is long, there will typically be many different paths for interest rates up to the target horizon that achieve the specified inflation target. If this is the case, a fixed inflation target at the specified horizon does not yield a unique value or time path for the interest rate, which is the actual instrument of monetary policy. A further special-ization for achieving the fixed target is to use inflation forecasts that are derived as constant interest rate (CIR) projections, see e.g. the discussions in (Leitemo 2003) and (Svensson 1999). CIR inflation targeting has been advocated as an easily understandable and hence practical approach to con-ducting monetary policy; for general discussions of its merits and problems see (Goodhart 2000), (Kohn 2000), (Svensson 2003b) and (Woodford 2003), pp. 620-623.

In practice there appear to be at least two different ways for computing and employing the CIR projections in setting the value for the monetary policy instrument. One approach, which is arguably close to the practice in the UK, has been described by (Goodhart 2000), p.177: ”When I was a member of the MPC I thought that I was trying, at each forecast round, to set the level of interest rates so that, without the need for future rate changes, prospective (forecast) inflation would on average equal the target at the policy”.1 _{Given a model of the macroeconomy, setting the forecast of}

inflation based on constant interest rates at a given target level of inflation implies a rule for the interest rate.

A second approach to CIR policy-making is in general terms described by the quote “... if the overall picture of inflation prospects (based on an unchanged repo rate) indicates that in twelve to twenty-four months’ time inflation will deviate from the target, then the repo rate should normally be adjusted accordingly”; see (Riksbanken 1999). This way of conducting monetary policy seems (at least implicitly) to be the practice in Sweden.2 In this approach the CIR projection is computed at the interest rate prevailing before any policy decision and the rate of interest is then adjusted depending on the difference between the CIR projection and the inflation target.

We will refer to these two ways of conducting monetary policy in general as CIR inflation targeting and corresponding interest rate rules as CIR rules. In addition, we will refer to the two approaches asCIRU K andCIRSpolicies,

respectively.

CIR inflation targeting necessarily introduces a further element of forward-looking behavior into the economy in addition to the forward-forward-looking be-havior of private agents that is assumed in many current models for mon-etary policy. If the model has forward-looking elements, issues of deter-minacy of rational expectations equilibria (REE) and their stability under (adaptive) learning have been raised in the recent literature. (Bullard and Mitra 2002) have derived constraints on the interest rate instrument (or Tay-lor) rules that achieve stability and determinacy in a standard New Keynesian model of monetary policy. (Evans and Honkapohja 2003a) and (Evans and Honkapohja 2003b) have shown that some standard ways for implement-ing optimal policy under discretion or commitment can lead to the diffi -culties of indeterminacy and instability under learning. They also propose expectations-based optimal rules to overcome these problems. (Evans and Honkapohja 2004) survey this literature and provide further references.

Our principal goal in this paper is to analyze CIR policies from the point of view of determinacy and stability under learning. We will study both

CIRU K and CIRS policies in these respects. We will argue that CIRU K

policies can very often lead to unpleasant outcomes, i.e. the resulting REE can exhibit both indeterminacy and instability under learning. CIRSpolicies

be followed if it leads to drastic policy changes. Our second interpretation of CIR policy has a more gradualist approach to changing interest rates.

2_{Anders Vredin pointed to us that in practice policy appears to respond to other aspects}

policies perform better with regard to both determinacy and stability under learning, but they are not always problem-free either. We also examine the (more realistic) policy of flexible inflation targeting and the consequences of inherent inertia in inflation and output for the indeterminacy and instability results.

2

The Framework

2.1

The Basic Model

The model we employ is the standard New Keynesian model of monopolistic competition and (Calvo 1983) price stickiness. This model has been employed in numerous recent studies of monetary policy; see e.g. (Clarida, Gali, and Gertler 1999) for a survey. The log-linearized model is described by two equations

xt=−ϕ(it−Et∗πt+1) +Et∗xt+1+gt, (2)

which is the “IS” curve derived from the Euler equation for consumer opti-mization, and

πt =λxt+βEt∗πt+1+ut, (3)

which is the price setting rule for the monopolistically competitive firms.3

We remark that in a later section we will add inertia terms to (2) and (3). The inertia is usually justified by empirical relevance even though the micro foundations of the model are then fairly weak.4

Herext and πt denote the output gap and inflation for period t,

respec-tively. it is the nominal interest rate, expressed as the deviation from the

steady state real interest rate. The determination of it will be discussed

be-low. E∗

txt+1 and Et∗πt+1 denote the private sector expectations of the output

gap and inflation next period. Since our focus is on learning behavior, these expectations need not be rational (Etwithout∗denotes RE). The parameters

ϕ and λ are positive andβ is the discount factor so that 0<β <1.

3_{See e.g. (Woodford 1996) or (Woodford 2003) for further details of the linearization}

and the original nonlinear model.

4_{See (Christiano, Eichenbaum, and Evans 2001) and (Gali and Gertler 1999) for}

The shocksgt and ut are assumed to be observable and follow µ gt ut ¶ =V µ gt−1 ut−1 ¶ + µ ˜ gt ˜ ut ¶ , (4) where V = µ µ 0 0 ρ ¶ ,

0 <_|µ_| <1, 0<_|ρ_|<1 and ˜gt ∼iid(0,σ2g), ˜ut ∼ iid(0,σ2u) are independent

white noise. gt represents shocks to government purchases and or potential

output. utrepresents any cost push shocks to marginal costs other than those

entering through xt. For simplicity, we assume throughout the paper that µ

and ρ are known (if not, they could be estimated).

For brevity, details of the derivation of equations (2) and (3) are not dis-cussed. The derivation is based on individual Euler equations under (identi-cal) subjective expectations, together with aggregation and definitions of the variables. The Euler equations for the current period give the decisions as functions of the expected state next period. Rules for forecasting the next period’s values of the state variables are the other ingredient in the descrip-tion of individual behavior. Given forecasts, agents are assumed to make decisions according to the Euler equations.5

For further analysis we write the model in matrix-vector form

yt = AEt∗yt+1+Bwt+Dit, (5)

wt = V wt−1+vt,

where yt = (xt,πt)0, wt = (gt, ut)0 and vt = (˜gt,u˜t)0. Et∗yt+1 denotes private

expectations of yt+1. The coefficient matrices are A= µ 1 ϕ λ β+λϕ ¶ , B = µ 1 0 λ 1 ¶ , D= µ −ϕ −λϕ ¶ . (6)

In the next two subsections we introduce the formalization ofCIRU K and

CIRS policies. In this and the next section, we consider the case when the

5_{This kind of behavior is boundedly rational but in our view reasonable since agents}

attempt to meet the margin of optimality between the current and the next period. Other models of bounded rationality are possible. Recently, (Preston 2002) has proposed a formulation in which long horizons matter in individual behavior. See also (Honkapohja, Mitra, and Evans 2002) for further discussion.

central bank tries to hit a certain inflation target at a specified horizon. Such a policy is often termed one of strict inflation targeting, following (Svensson 1999) and (Svensson 2003a). Even though this policy is not entirely realistic from a practical point of view since most central banks have output concerns (either implicitly or explicitly), it does serve as a useful benchmark. The case when the bank pursues a policy of flexible inflation targeting is considered in Section 4.1.

2.2

CIR

Policy

We first consider CIRU K formulation of CIR policy. This has been recently

formalized by (Leitemo 2003), which can be consulted for further details. We introduce the (strict) inflation target as in (1), where for simplicity the target ¯π is assumed to be zero without loss of generality (w.l.o.g.) and h is the targeting horizon.6

In the derivation of CIR policies it is assumed that the central bank acts as if expectations of private agents are rational and the forecasts are computed under knowledge of the structural model of the economy. For later purposes it will be useful to express this constraint as

0 =K(Etwt0+h, Ety0t+h)0, K = (0,0,0,1). (7)

To derive the interest rate rule, rewrite (5) as

µ wt+1 Etyt+1 ¶ =Ω µ wt yt ¶ +Ψit+ µ vt+1 0 ¶ , (8) where Ω = µ V 0 −A−1_{B A}−1 ¶ ≡     µ 0 0 0 0 ρ 0 0 −1 ϕβ−1 1 +λϕβ−1 ₋ϕβ−1 0 ₋β−1 ₋λβ−1 β−1    , Ψ = µ 0 −A−1_D ¶ ≡( 0 0 ϕ 0 )0.

6_{This is without loss of generality as the precise values of model constants affect neither}

Iterate (8) forward to get µ Etwt+h Etyt+h ¶ =Ωh µ wt yt ¶ + h−1 X j=0 ΩjΨEtit+h−1−j, (9) Pre-multiplying (9) by K yields Etπt+h =KΩh µ wt yt ¶ +K h−1 X j=0 ΩjΨEtit+h−1−j. (10)

CIRU K targeting policy central bank assumes that

Etit+j =it for 0≤j ≤h−1, (11)

where in (11) it is assumed that expected future interest rates are equal to the contemporaneous interest rate it for all horizons 0≤j ≤h−1.In other

words, in the formulation of policy, the bank assumes a constant path of interest rates at the current level. Using assumption (11) in (10) leads to

Etπt+h(it) =KΩh µ wt yt ¶ +K h−1 X j=0 ΩjΨit (12)

where Etπt+h(it) denotes the constant-interest-rate forecast of inflation

con-ditional on the forward-looking variables xt,πt and the contemporaneous

interest rate it. Finally, setting Etπt+h(it) in (12) equal to the (target) zero

yields the interest rate rule

it =G µ wt yt ¶ ,G=₋ Ã K h−1 X j=0 ΩjΨ !−1 KΩh. (13)

We will refer to (13) as the CIRU K rule I.

The CIRU K rule I, equation (13), has the general form

it =χggt+χuut+χxxt+χππt. (14)

(14) is thus an instrument rule like the classic rule studied by (Taylor 1993) and it can be explicitly computed for different values of h. For h= 2 we get

χg =ϕ−1,χu =− 1 +βρ+λϕ βλϕ ,χx =− 1 +β+λϕ βϕ ,χπ = 1 +λϕ βλϕ .

It is seen that, for h = 2, the rule (13) surprisingly has χx < 0, i.e. the

interest rate should react negatively to the output gap. For higher values of h the expressions χ_i, i = g, u, x,π become cumbersome, but numerical computations indicate that the negative coefficient on the output gap is a robust phenomenon of the CIRU K rule I.

This unexpected result can be given an economic interpretation in the case h= 2. Shift the New Phillips curve (3) forward and take RE. Recalling that the inflation target is assumed to be zero, we have

Etπt+1 =λEtxt+1+ρut,

which pins down the expectations terms in (2) and yields the positive relation between Etπt+1 and Etxt+1. By (3) we also have

Etπt+1 =β−1(πt−λxt−ut),

which indicates that bothEtπt+1andEtxt+1depend negatively on the current

output gap under this policy. Finally, rewriting the IS curve (2) as

ϕit = −xt+ϕEtπt+1+Etxt+1+gt

= ₋(1 +β−1+β−1λϕ)xt+ (β−1ϕ+β−1λ−1)πt

+gt−(β−1ϕ+λ−1(β−1+ρ))ut.

it is seen that itand xtare negatively related, both directly as part of the IS

relationship and indirectly through the negative dependence of Etπt+1 and Etxt+1 on the current xt.

(13) should be viewed as an instrument rule as it depends on current endogenous variables7 _{and another rule depending only on predetermined}

variables is often suggested instead. It can be derived as follows. Substituting (13) into (8) we have µ wt+1 Etyt+1 ¶ = (Ω+ΨG) µ wt yt ¶ + µ vt+1 0 ¶ ,

for which it is possible to derive the MSV solution of the form

yt=Hwt (15)

7_{(13) can be viewed as a behavioral rule in the same sense as demand and supply}

functions of private agents are behavioral rules, i.e. the central bank “goes to the market” with that schedule and is able to adjust the interest rate within the period. Such rules are sometimes said to be non-operational; see (McCallum 1999) for further discussion.

using standard techniques (we omit the precise form of H). (The MSV solutions are REE that are usually employed in the applied literature.) In-troducing the partition G = ¡ Gw Gy

¢

, we can rewrite the interest rule (13) as

it= (Gw+GyH)wt, (16)

which we will call the CIRU K rule II.

2.3

CIR

Policy

As mentioned in the Introduction, an alternative interest rule, which we call aCIRS rule, is based on computing CIR forecasts of inflation at the interest

rate before any policy decision and then changing the interest rate if the CIR forecast deviates from the target. Formally, the policy-maker first makes a forecast of inflation, conditioned on a constant interest rate at the level of

it−1 from period t−1. This forecast is compared with the target rate. If the

forecast is above the target, the interest rate is raised. One simple rule that reflects this way of thinking is8

it−it−1 =ω(Etπt+h(it−1)−π¯), (17)

where Etπt+h(it−1) denotes the inflation forecast conditioned on the last

pe-riod interest rate, i.e., the bank assumes

Etit+j =it−1, for all 0≤j ≤h−1

instead of (11). We again assume ¯π = 0, w.l.o.g. ω > 0 determines the magnitude of the extent of increase in it when the inflation forecast is above

target. From (10) we now have

Etπt+h(it−1) =KΩh µ wt yt ¶ +K h−1 X j=0 ΩjΨit−1 (18)

Written explicitly, (18) takes the form

Etπt+h(it−1) =ψggt+ψuut+ψxxt+ψππt+ψiit−1 (19)

8_{This is a simplified version of a rule proposed by Anders Vredin in the discussion at}

where the coefficientsψg,ψu,ψx,ψπ,ψi can be computed for different values

of h. For example, when h= 2,these take the values

ψ_g = λβ−1,ψ_u =₋β−2(1 +βρ+λϕ),ψ_x =₋λβ−2(1 +β+λϕ), ψ_π = β−2(1 +λϕ),ψ_i =₋λϕβ−1.

Using (19) in (17), we obtain CIRS rule I of the form

it=ω(ψggt+ψuut+ψxxt+ψππt) + (1 +ωψi)it−1 (20)

Note that the CIRS rule I, (20), captures a form of interest smoothing

fre-quently observed in the data.

As before, we can also define an interest rule which depends solely on pre-determined variables using the MSV solution of the model when CIRS

rule I is employed. We now consider the formulation of the CIRS rule II

associated with (17). Since the CIRS rule I introduces the lagged interest

rate as a predetermined endogenous variable, the MSV solution of the model (5) with CIRS rule I, (20), takes the form

xt = bxit−1+bgxgt+buxut, (21)

πt = bπit−1+bgπgt+buπut, (22)

it = biit−1+bgigt+buiut, (23)

where the coefficients bx, ... need to be determined. Furthermore, for a

de-terminate MSV solution, we have _|bi| < 1, and using this solution in CIRS

rule I, (20), we may obtain uniquely CIRS rule II below9

it=b0iit−1+ ˜ψggt+ ˜ψuut. (24)

Hereb0

i =ωψxbx+ωψπbπ+ (1 +ωψi) and ˜ψg,ψ˜u describe the dependence on

the shocks (their precise form is not needed in the computations).

2.4

Calibration Scenarios

In several places we will need to revert to numerical results in the study the properties of CIR policies introduced above. For our numerical analysis, we will frequently adopt three calibration scenarios proposed in the literature.10

9_{In cases when the model has indeterminacy and hence potentially multiple stationary}

MSV solutions withCIRS rule I, there is no unique way to defineCIRS rule II.

10_{Both the (Clarida, Gali, and Gertler 2000) and (Woodford 1999) calibrations are}

Calibration W: β = 0.99,ϕ= (0.157)−1_{, and λ= 0.024.} Calibration CGG: β = 0.99,ϕ= 4, and λ= 0.075. Calibration MN: β= 0.99, ϕ= 0.164, and λ= 0.3.

These are taken, respectively, from (Woodford 1999), (Clarida, Gali, and Gertler 2000), and (McCallum and Nelson 1999). We remark that these calibrations are based on U.S. data and thus the numerical results are not necessarily relevant for the British and Swedish cases. For an analysis of E-stability, we sometimes need the values of ρ and µ and we set these at

ρ= 0.9 andµ= 0.35 in accordance with the literature.

3

Determinacy and Learning Stability

3.1

Results for

CIR

Policies, Types I and II

We consider whether CIRU K interest rate rules, either in the form (13) or

(16), yield determinacy and stability under learning of the MSV REE. We will assess stability under learning using the concept of E-stability, which is known to be the relevant condition for convergence of adaptive learning formulated using least squares and closely related learning rules. Formal analysis of determinacy is standard; see e.g. (Blanchard and Kahn 1980) or Chapter 10 of (Evans and Honkapohja 2001). For an analysis of E-stability in models like this, we refer the reader to (Bullard and Mitra 2002) for an overview and to (Evans and Honkapohja 2001) for a detailed discussion. The analysis is conducted using the forward-looking model (2) and (3), together with either (13) or (16).

It should be emphasized that the analysis of determinacy and learning is based on the assumption that the private sector expectations are based on expectation functions with the actually implemented interest rate rule and not on expectations used as part of the computation of the hypothetical CIR policy. This highlights an aspect of the time-inconsistency of CIR policy: computation of the policy presumes a constant interest rate through the

inflation as quarterly changes in the log price level, while (Clarida, Gali, and Gertler 2000) use annualized rates for both variables. We adopt the Woodford measurement convention, and therefore our CGG calibration divides by 4 theλvalue and multiplies by 4 theϕvalue reported by (Clarida, Gali, and Gertler 2000).

horizon, whereas the actual interest rate is known to be adjusted in response to variations in inflation and output gap as well as in the shocks.11

It is convenient to start withCIRU K rule II (16). Plugging this rule into

the model (5) leads to the reduced form

yt =AEt∗yt+1+{B +D(Gw+GyH)}wt. (25)

In the basic model (16) has unpleasant properties on both counts:12

Proposition 1 CIRU K rule II, i.e. equation (16), leads to both

indetermi-nacy and instability under adaptive learning.

The indeterminacy result means that there other stationary REE to the model under theCIRU K rule II besides the MSV solution used above. These

equilibria include various sunspot solutions and it is possible to examine whether the non-MSV solutions are stable under learning. Using results of (Honkapohja and Mitra 2004), it can be shown that the non-MSV REE are also E-unstable. Thus there are no E-stable REE under CIRU K rule II.

The difficulties spelled out by Proposition 1 naturally raise the question whether the instrument rule form of CIR monetary policy, i.e. CIRU K rule

I given by equation (13) has better determinacy or learnability properties. Unfortunately, this is not the case:

Proposition 2 In model (2)-(3)CIRU Krule I leads to indeterminacy (when

h_≥2) and to E-instability (when h_≥3).

In Appendix A we prove the result for values h _≤ 4. For higher values of h we have computed the relevant conditions numerically using the three baseline calibrations. The results clearly indicate that the CIRU K rule I

delivers neither determinacy nor stability under learning.

3.2

Results for

CIR

Policies, Types I and II

We now turn to an analysis of the performance of CIRS type policy. To

analyze determinacy, we plug the rule (20) into the basic model (5) and

11_{See (Leitemo 2003) for a further discussion of time-inconsistency issues in CIR policies.} 12_{The result follows directly from Proposition 2 in (Evans and Honkapohja 2003a)}

stat-ing that, in the New Keynesian model, any interest rate rule that depends only on the exogenous shocks lead to both indeterminacy and instability under learning.

obtain the following system after defining zt= (xt,πt, it−1)0 B1Etzt+1 =B2zt+shocks, (26) where B1 =   1λ β+ϕλϕ 00 0 0 1  , B2 =   1 +λϕωψϕωψ_xx 1 +ϕωψλϕωψπ _π λϕϕ(1 +(1 +ωψωψi)_i) ωψ_x ωψ_π 1 +ωψ_i  

The matrix for computing determinacy is then given byB3 ≡B1−1B2, which

explicitly is B3 =   1 +β −1_λϕ+ϕωψ x β− 1_ϕ(βωψ π −1) ϕ(1 +ωψi) −β−1λ β−1 0 ωψx ωψπ 1 +ωψi  

Since xt,πt are free whileit−1 is pre-determined, REE is determinate if and

only if exactly one eigenvalue of B3 is inside the unit circle.

Whenh= 2,we can obtain a partial result on determinacy. The following proposition is proved in Appendix B:

Proposition 3 Let h = 2. CIRS rule I yields determinacy of REE for all

sufficiently small ω >0.

In general, determinacy depends on the structural parameters. It is easily checked that in the case h = 2 determinacy obtains for all 0 < ω < 1 under the three baseline calibrations. For higher horizons we will examine determinacy numerically below.

To analyze E-stability, we need to put the system in the following form

ςt = zEˆtςt+1+δςt−1+κwt, (27) z = %   1λ β+ϕ(1λϕ−+ωβψβϕωψπ) π 00 ω(λψ_π+ψ_x) ω_{(β+λϕ)ψ_π +ϕψ_x} 0  , δ = %   0 00 0 ₋−λϕϕ(1 +(1 +ωψωψi)i) 0 0 1 +ωψ_i  , %−1 = 1 +ϕω(λψπ+ψx).

where ςt= (xt,πt, it)0 and ψx,ψπ,ψi are the coefficients in (19).

The MSV solution of the model (27) takes the form

ςt= ¯a+ ¯bςt−1+ ¯cwt (28)

with ¯a = 0 and ¯b to be determined from ¯b = (I _{− z}¯b)−1_{δ, provided the}

relevant inverse exists. In the determinate case, there is only one solution of ¯

bwith eigenvalues inside the unit circle; in the indeterminate case there may exist more than one.

For the analysis of learning, agents have a PLM of the form

ςt=a+bςt−1+cwt,

from which one can compute expectations13 _{as ˆE}

tςt+1 = (I+b)a+b2ςt−1+

(bc+cV)wt and inserting ˆEtςt+1 into the model gives the ALM ςt = (z+zb)a+ (zb2+b)ςt−1+ (zbc+zcV +κ)wt

When the time t information set is (1,ς0t−1, wt)0, the E-stability conditions

for an MSV solution require us to have the eigenvalues of the matrices ¯b0_⊗

z+I_⊗z¯b₋I, V _⊗z+I_⊗z¯b₋I,_z+_z¯b₋I, to have negative real parts; see Chapter 10 of (Evans and Honkapohja 2001) for details. Otherwise, the solution is not E-stable.

We now look numerically at E-stability of the determinate MSV solution for different horizonsh, ranging from 2 to 8. Table 1 below reports a pair of critical values of ω, (ω,ˇ ωˆ) such that for all 0<ω _≤ω,ˇ one has determinacy with the determinate MSV solution being E-stable. For values of ω such that ωˇ <ω _≤ω,ˆ one has determinacy but the determinate MSV solution is E-unstable. Values of ω > ωˆ lead to indeterminacy.14 _{For h = 2, we find}

that ωˇ = ˆω so that only one number is reported for the h= 2 column.

Table 1. Regions of Determinacy and E-stability for different horizons

h 2 3 4 5 6 7 8

W 3.39 (.84, .84) (.3, .34) (.14, .17) (.07, .1) (.04, .06) (.03, .03) CGG 1.87 (.44, .46) (.14, .18) (.06, .09) (.03, .04) (.01, .02) (.01, .01) MN 10.39 (2.4,2.4) (.94, .94) (.45, .47) (.25, .28) (.15, .18) (.10, .12)

13_{We assume that the timetinformation set does not includeς} t.

14_{We have not conducted an analysis of E-stability in the indeterminate region. We did}

The determinate solution usually turns out to be E-stable (there are only some exceptions). However, the range for which determinacy and E-stability hold shrinks quite rapidly as the horizon increases. When h = 8, only very small values of ω yield determinacy and E-stability.

Note that, in the interest rule (20), the responses to xt, πt, and it−1 are,

respectively, ωψ_x,ωψ_π, and 1 +ωψ_i and these resemble the form of a Taylor type rule with interest rate smoothing. With this rule, the Taylor principle (T P) corresponds to the requirement

T P _≡1 +ωψ_i+ωψ_π+λ−1(1₋β)ωψ_x >1; (29) see Chapter 4 of (Woodford 2003) for a discussion of the Taylor principle. It can be verified analytically that for all horizons h = 2,3, ..,8, T P = 1 +ω

so that the rule does indeed satisfy the Taylor principle for all ω > 0. In fact, Proposition 4.4, p. 255, of (Woodford 2003), shows that the necessary and sufficient condition for determinacy, for a rule of the form (20), in the model (5) is that the Taylor principle be satisfied, i.e. that T P >1provided

ψx > 0,ψπ > 0, and 1 +ωψi > 0. The latter means that the individual

responses to xt,πt, and it−1 are all positive (as is perhaps true in realistic

versions of the Taylor type rule). However, for the rule (20), ψ_x <0 for all calibrations and often 1 +ωψi <0,(even thoughψπ >0 for all calibrations).

We conjecture that the key to the failure of determinacy and E-stability is the fact that the rule (20) fails to be super-inertial since 1 +ωψ_i <1.With strict inflation targeting, it can be verified analytically that ψi < 0 for all

horizons h so that the rule can merely be inertial.15 _{An explanation for this}

conjecture will be provided later.

ForCIRS rule II we have both indeterminacy and E-instability (the proof

is in Appendix B):

Proposition 4 CIRS-rule II associated with (19), i.e. equation (24), leads

to both indeterminacy and instability under adaptive learning of the MSV solution for all horizons h.

15_{In fact, as ω increases, the response 1 +ωψ}

i becomes an increasingly large negative

4

Extensions

4.1

Flexible In

fl

ation Targeting

We examine some extensions to our basic model. The analysis of Section 2 has assumed that the central bank pursues a policy of strict inflation target-ing, which serves as a useful benchmark. We now turn to the arguably more realistic case where the bank also has concerns for output in its loss function. With flexible inflation targeting and assuming that the target levels for output gap and inflation are ¯xand ¯π, the central bank’s optimality condition (under discretion) can be shown to beαxt+λπt−(λ¯π+αx¯) = 0,see (Evans

and Honkapohja 2003a) for the details.16 Thus, the bank seeks to achieve

αλ−1Etxt+h+Etπt+h−(¯π+αλ−1x¯) = 0. (30)

We again rewrite this constraint as

¯

π+αλ−1x¯=K(Etwt+h, Etyt+h)0, K = (0,0,αλ−1,1). (31)

Except for the change in K, the rest of the analysis formally proceeds as before. The form of CIRU K rule I continues to be given by (13) with K as

defined in (31). As before, we assume ¯x = ¯π= 0 (w.l.o.g.).

Obviously, some assumptions about α must be made. In the numerical analysis we assumed that α ranges from 0.1 (low concern for output) to 0.9 (high concern for output) at intervals of 0.1. We continue to have:17

Result: Under CIRU K rule I the REE is indeterminate and the MSV

solu-tion is E-unstable.

This result was obtained for all the examined α and across all three calibra-tions. As for CIRU K rule II, Proposition 1 continues to be applicable for

the case offlexible inflation targeting since the rule still depends only on the exogenous shocks.

16_{(Evans and Honkapohja 2003a) consider optimal discretionary policy by minimizing a}

quadratic objective function,α(xt−x¯)2+ (πt−¯π)2,subject to (3), which approach leads

to the discretionary optimality condition.

4.1.2 CIRS Rules

Since the bank’s targeting rule is now given by (31), a rule analogous to (17) in this case can be written as

it−it−1 =ω[αλ−1Etxt+h(it−1) +Etπt+h(it−1)−(¯π+αλ−1x¯)] (32)

If the bank expects the expression within parentheses in (32) to be positive, then interest rates should be raised to reduce inflationary pressures in the economy. According to (32) the nominal interest rate is raised if the optimal combination of forecasted output gap and inflation exceeds the corresponding combination evaluated at the target values.18

For the formal analysis we remark that αλ−1Etxt+h(it−1) +Etπt+h(it−1)

will be of the same form as the right hand side of (18) except for the change inK, namely K = (0,0,αλ−1,1).Again, we simplify by assuming ¯x= ¯π = 0 and continue to obtain from (32), an interest rule of the form (20) but with different coefficients. For example, withh= 2, the coefficients are

ψx = λ− 1 β−2[α_{β2+ (1 + 2β)λϕ+λ2ϕ2_}−λ2(1 +β+λϕ)], ψπ = λ− 1 β−2[λ(1 +λϕ)₋αϕ(1 +β+λϕ)], ψi = ϕλ− 1 β−1[α(2β+λϕ)₋λ2].

Note that for large enough α, ψi > 0 (and ψx >0,ψπ <0). One can verify

analytically for higher horizons that the same features are true and, conse-quently, the interest rules are super-inertial for α large enough.

It can also be verified analytically that for all horizons h = 2,3, ..,8 the interest rule (20), continues to satisfy the Taylor principle since

T P = 1 +ω+λ−2αω(1₋β)>1 for all α,ω >0.

Since onlyK changes, the rest of the formal analysis proceeds as in Sec-tion 2.3. We examine determinacy and E-stability forCIRS rule I for values

of 0 < ω _≤ 15 and for values of α between 0.1 and 0.9, both at intervals of 0.1. Remarkably, numerical results suggest the following general conclusion:

18_{Responding to deviations from the optimality condition is similar in spirit to the}

approximate targeting rule proposed by (McCallum and Nelson 2000). However, in the McCallum-Nelson rule a deviation from optimality leads to an increase in the real interest rate.

Result: Most values of α lead to determinacy and E-stability of the deter-minate solution for horizons h= 2,3, ..,8.

The conclusion shows that if the bank has sufficient concerns for output in its loss function and adopts a rule of the form (32), CIRS rule I policy performs

well in terms of determinacy and E-stability.

The detailedfindings are as follows. For the W and CGG calibrations, we

find that for allh,all values ofα,ω examined lead to determinate REE which are also E-stable. The corresponding rule (20) has its coefficients satisfying

ψ_x > 0, ψ_π < 0, and 1 +ωψ_i > 1. So even though the response to πt is of

the ”wrong” sign, the rule is nevertheless super-inertial.

For the MN calibration, the general theme is unchanged. We find that values of α _≥ 0.3 lead to determinacy and E-stability for all h. In other words, a sufficient concern for output eliminates problems of determinacy and E-stability. In addition, the associated rules tend to be super-inertial since 1 +ωψ_i >1, i.e.,ψ_i >0 (they also satisfy ψ_x >0 andψ_π <0).Determinacy sometimes fails for small values ofαlikeα=.1, .2 when the horizonhis large (say, h _≥ 4). These failures of determinacy typically coincide with interest rules which satisfy 1 +ωψ_i <1 (along withψ_x <0,ψ_π >0),i.e. when policy rules that are not super-inertial.

We note that the MSV solution has bx < 0, bπ < 0, and 0 < bi < 1

for all calibrations. These results together with those of the previous section suggest that an important reason for determinacy and E-stability is the super-inertial nature of the associated interest rule, which seems to be true even when ψ_π < 0, or ψ_x < 0.19 _{(Bullard and Mitra 2001) examined}

super-inertial interest rules (with ψπ >0,ψx>0 and dependence on lagged data)

and found these to be conducive to E-stability of the MSV solution for the basic model (5). They also found that superinertial rules that depend on contemporaneous data on inflation and output and the lagged interest rate, as in rule (20), were conducive to determinacy and E-stability.

Finally, we remark that Proposition 4 continues to be applicable forCIRS

rule II since it was applicable for any determinate MSV solution under all parameter values.

19_{For some (intermediate) values ofα, the individual responses in the interest rule to}

output, inflation and lagged interest rates are all positive (as with the MN calibration) and since the rule also always satisfies the Taylor principle, the determinacy result in Proposition 4.4 of (Woodford 2003) is applicable.

4.2

In

fl

ation and Output Inertia

The model given by (2) and (3) is entirely forward-looking and as a result has difficulty capturing the inertia in output and inflation evident in the data; see (Fuhrer and Moore 1995b), (Fuhrer and Moore 1995a) and (Rudebusch and Svensson 1999) for empirical results. We now look at an extension of this model considered in (Clarida, Gali, and Gertler 1999), Section 6, with important backward-looking elements. This model consists of the structural equations

xt = −ϕ(it−Et∗πt+1) +θEt∗xt+1+ (1−θ)xt−1+gt (33)

πt = λxt+βγEt∗πt+1+ (1−γ)πt−1 +ut (34)

The parameters θ and γ capture the inertia in output and inflation inherent in the model and are assumed to be between 0 and 1. The shocksgt and ut

continue to follow the process (4).

We outline the formal analytical procedures in Appendix C.

4.2.1 CIRU K Rules

The CIRU K rule I, equation (42), under strict inflation targeting has the

general form

it =ϑggt+ϑuut+ϑxLxt−1+ϑπLπt−1+ϑxxt+ϑππt (35)

and it is possible to compute this rule explicitly for different values ofh. For instance, with h= 2, the rule is

ϑg = ϕ−1,ϑu =− θ+βγθµ+λϕ βγλϕ ,ϑxL= 1₋θ ϕ ,ϑπL=− (1₋γ)(θ+λϕ) βγλϕ , ϑx = − θ+βγ+λϕ βγϕ ,ϑπ = θ_{1₋βγ(1₋γ)_}+λϕ βγλϕ . (36)

Note that the response of the interest rule to the contemporaneous output gap is negative (as in the non-inertial model) and, in addition, the response to lagged inflation is now negative. Similar qualitative responses follow for other horizons.

We examine determinacy and E-stability in the model with inertia (33), (34) for theCIRU K rule I (35) when the central bank pursues strict inflation

analyze this case first.20 _{When h= 2,there is indeterminacy for all values of}

output and inflation inertia. The MSV solution turns out to be unique but it is not E-stable:

Proposition 5 CIRU K rule I leads to indeterminacy in the model (39) when

h= 2. There exists a unique MSV solution, which is E-unstable.

The result is proved in Appendix C. For h > 2, we need to resort to numerical analysis and we letγ and θ take values from 0.1 to 0.9 at intervals of 0.1.Table 2 below reports the results whenh= 4 for the W calibration. In this table, the third column shows determinacy (D) or indeterminacy (I). The fourth column shows the number of stationary MSV solutions. Obviously, in the determinate case, there is only one stationary solution whereas there may be more than one in the indeterminate region. The final column examines E-stability of the stationary MSV solutions whether in the determinate or indeterminate region.

Table 2. CIRU K Rule I: E-stability of MSV solution when h=4

20_{For E-stability, we continue to assume that agents’s expectations are based on}

in-formation of endogenous variables at time t₋1, which we believe is more realistic since contemporaneous data on output and inflation are not usually available for making fore-casts.

γ θ Det/Indet # of stat solns E-stability .1 _{.1,..,.5_} D 1 No .1 _{.6,..,.9_} I 2 No in both cases .2 .1 D 1 Yes .2 _{.2,..,.5_} D 1 No .2 _{.6,..,.9_} I 2 No in both cases .3 .1 D 1 Yes .3 .2,.3,.4 D 1 No .3 _{.5,..,.9_} I 2 No in both cases .4 .1 D 1 Yes .4 .2,.3 D 1 No .4 _{.4,..,.9_} I 2 No in both cases .5 .1 D 1 Yes .5 .2 D 1 No .5 _{.3,..,.9_} I 2 No in both cases {.6,.7_{} {}.1,..,.9_} I 2 No in both cases .8 _{.1,..,.6_},.9 I 2 No in both cases .8 .7,.8 I 3 No in all cases .9 _{.1,..,.5_},.8 I 2 No in both cases .9 .6,.7,.9 I 3 No in all cases

The table shows that most values of inflation and output inertia lead to indeterminacy whenh= 4.Even if determinacy obtains, the (locally) unique solution is usually E-unstable. In the indeterminate region, all MSV solutions always turn out to be E-unstable.

Similar results follow for higher horizons. These results indicate that policy using CIRU K rule I continues to have undesirable properties in the

presence of inertia.

For brevity, we do not report the performance of CIRU K rule II here.

We have checked numerically that the qualitative features of this rule are basically unchanged from those of CIRU K rule I. Most parameter values

continue to lead to indeterminacy and all MSV solutions are E-unstable.

4.2.2 CIRS Rules

We consider the performance of CIRS rule I in the presence of flexible

presence of inflation inertia, the targeting rule (31) takes the form

0 =K(Ety1,t+h, Ety2,t+h)0, K = (0,0,0,0,α(1−β˜aπ)λ−1,1), (37)

compare (6.4) in (Clarida, Gali, and Gertler 1999). 0 _≤ ˜aπ < 1 is the

solution of the lagged inflation term in equation (6.5) of (Clarida, Gali, and Gertler 1999). When γ = 1, a˜π = 0. Wefirst summarize the general nature

of the results in this case.

Results: The CIRS rule I continues to perform well in the presence of low

levels of output and inflation inertia. The presence of high levels of inertia hampers the performance of this rule.

Consider first determinacy under the rule. Table 3 depicts the region of determinacy when h = 8 for the W and CGG calibrations. For each value of γ in the first row, the table reports the critical value ¯θ such that one has determinacy for all θ _≥ ¯θ for all values of α and ω examined.21 There typically exists no stationary REE for values of θ <¯θ.

Table 3. Region of Determinacy when h = 8 γ .1 .2 .3 .4 .5 _≥.6

W .6 .6 .5 .4 .3 .1 CGG .6 .5 .4 .3 .2 .1

It seems, therefore, that determinacy can fail in the presence of high levels of inertia in the model. In addition, even when determinacy holds, the (determinate) solution can fail to be E-stable.22 _{Thus, sufficient inertia}

worsens significantly the performance of CIRS rule I.

5

Concluding Remarks

The results in this paper suggest that the conduct of inflation targeting by us-ing CIR policy is subject to two fundamental difficulties. First, there may be multiple stationary RE solutions under such a policy. Second, the suggested

21_{We examined determinacy for values of γ,θ,αbetween.1 and.9 and for values ofω}

between.1 and 2,all at intervals of length.1.

22_{For simplicity, we considered E-stability in the presence of inflation inertia only, i.e.,}

interest rates rules in this approach can lead to instability of equilibria under learning. We remark that optimal inflation targeting policies, discussed e.g. in (Svensson 2003a), are an alternative to CIR policies and there are ways to implement them to achieve determinacy and learnability; see (Evans and Honkapohja 2004).

We have examined two versions of CIR inflation targeting, which we called

CIRU KandCIRS.It was found thatCIRU Kpolicies are particularly

vulner-able to the twin problems of indeterminacy and E-instability in all versions of the models examined. CIRS rule I, on the other hand, has more

appeal-ing features in terms of determinacy and E-stability in the forward-lookappeal-ing model, especially when flexible inflation targeting is employed. However, its performance can be problematic in the presence of high inflation or output inertia. One reason for the poor performance may be the relative simplicity of the rule itself. In inertial models one may have to look at other rules to deliver a robust performance. We leave a detailed investigation of these issues to the future.

It is perhaps remarkable that the performance of CIRS instrument rule

in terms of learnability and determinacy is superior to CIRU K policy, which

is an optimal policy under RE. The response coefficientωmakesCIRS more

flexible thanCIRU K policy. In fact, CIRU K rule I is a special case of CIRS

rule I when ω=₋ψ−_i 1, which is easily verified comparing equation (13) with (17)-(19).

The basic analysis can be extended in various ways. First, we made the strong assumption that the central bank knows the structural parameters of the economy when it computes the CIR interest rate rule. If structural parameters are not known, they can be estimated from data as in (Evans and Honkapohja 2003a) and (Evans and Honkapohja 2004). A result of (Evans and Honkapohja 2003a) shows that an interest rate rule that leads to instability under learning when the policy-maker knows the structural parameters does not fare better when structural parameters are estimated. We conjecture that an analogous result will hold for CIR interest rate rules. Second, we have limited attention to the computation of CIR policy sug-gested by (Leitemo 2003). While Leitemo’s approach is very natural, the appendix of (Svensson 1998), which is the unpublished version of (Svensson 1999), suggests a different formulation of what is meant by inflation targeting with a fixed target at fixed horizon. Svensson’s approach is quite general as he constructs consistent internal forecasts relative to any fixed interest rate rule beyond a specified horizon. However, formulations of Svensson’s

ap-proach in the basic forward-looking model make the interest rate dependent only on exogenous shocks, and a result analogous to Proposition 1 is then applicable. Moreover, as pointed out by (Leitemo 2003), further consistency restrictions naturally arise. For example, Leitemo’s rules of the form (13) and (16) do not meet consistency beyond and within the targeting horizon.

Appendices

A

Proof of Proposition 2

To prove Proposition 2 we first note that it is unnecessary to consider the exogenous shocks for these results. They play no role in indeterminacy and also, in this setting, E-instability also follows from considering the model without the shocks. We first consider determinacy for 4 _≥ h _≥ 2 and E-stability for 4_≥h >2.

Substituting (14) into (5) and omitting the shocks, we have the system

M yt = N Et∗yt+1, where M = µ 1 +ϕχ_x ϕχ_π −λ 1 ¶ , N = µ 1 ϕ 0 β ¶ .

Since both variables are free, we need both eigenvalues of M−1_{N to be}

in-side the unit circle for determinacy whereas for E-stability we need the real parts of the eigenvalues of M−1_{N to be less than 1. Alternatively,}

condi-tions for determinacy and E-stability may be given in terms of the trace and determinant of M−1N as the system is two-dimensional.

The necessary and sufficient condition for determinacy turns out to be

0 > Abs[Det(M−1N)]₋1

0 > Abs[T r(M−1N)]₋1₋Det(M−1N),

where Abs refers to the absolute value of the bracketed expression. The necessary and sufficient condition for E-stability turns out to be

T r(M−1N ₋I) < 0, Det(M−1N ₋I) > 0.

We now examine determinacy and E-stability for various horizonshusing the above conditions. In the case h = 2 matrix M is singular, but we can assess determinacy by computing

N−1M = µ −β−1 (βλ)−1 −λβ−1 β−1 ¶ .

Both eigenvalues of N−1_{M are zero, so that we have indeterminacy.}23

In the caseh= 3 we get

M−1N = Ã _1+2β+λϕ β −1+(β−1)λϕ βλ λ(1+2β+λϕ) β (β−1)(1+β+λϕ) β ! .

It is easy to check thatDet(M−1_N)

−1 = 2β+λϕ>0, so that indeterminacy prevails. In addition, T r(M−1_N

−I) =β+λϕ>0 implying E-instability. In the caseh= 4 we have

Det(M−1N)₋1 = 3β 2_{+ 3βλϕ+λϕ(1 +λϕ)} 1 + 2β+λϕ >0, T r(M−1N ₋I) = 2β 2 + 3βλϕ+λϕ(1 +λϕ) 1 + 2β+λϕ >0

so that both indeterminacy and E-instability prevail.

B

Results for

CIR

Rules

Proof of Proposition 3: When h= 2,the characteristic polynomial of the determinacy matrix B3 is

p(τ) _≡ τ3+C2τ2+C1τ +C0,

C2 = −β−2(1 + 2β+λϕ)(β−λϕω), C1 = β−2(2 +β+λϕ)(β−λϕω), C0 = −β−2(β−λϕω).

Then computing p(1) = 1 +C2 +C1+C0 and p(−1) = −1 +C2 −C1 +C0

one obtains

p(1) = β−1λϕω>0,

p(₋1) = ₋β−2[4β2 +β(4 +λϕ(2₋3ω))₋2λϕω(2 +λϕ)].

23_{We remark that the analysis of E-stability forh= 2 is not considered as the singularity}

Woodford has given necessary and sufficient conditions for exactly one eigen-value of B3 to be inside the unit circle (which is the condition for

determi-nacy) in terms of the characteristic polynomial, see Proposition C.2, p. 672 of (Woodford 2003). Since p(1) > 0, we are in Woodford’s Cases II or III. When ω >0 is sufficiently small we have p(₋1)<0, which leads to his Case III. Forω >0 is sufficiently small it is easily verified that_|C2|>3 and so we

get determinacy.

Proof of Proposition 4: Using (24) in the basic model (5), we obtain the following system, where ςt = (xt,πt, it)0,

ςt = zˆEˆtςt+1+ ˆδςt−1+ ˆκwt, (38) ˆ z =   λ β1 +ϕλϕ 00 0 0 0  ,ˆδ=   0 0 −ϕb 0 i 0 0 ₋λϕb0 i 0 0 b0 i  .

We note that the form of the MSV solution with CIRS rule II takes the

same form as (21)-(23), except thatbx, bπ, bi (and also other coefficients) take

different values. For future reference, we denote these values respectively by

b0

x, b0π, b0i.

Definingzt= (xt,πt, it−1)0, for determinacy we need to look at the system

B1Etzt+1 = ˆB2zt; ˆB2 =   1 0 ϕb 0 i 0 1 λϕb0 i 0 0 b0 i  

and B1 defined in (26). REE is determinate iff the matrix

B₁−1Bˆ2 =   1 +λϕβ −1 −ϕβ−1 ϕb0 i −λβ−1 β−1 0 0 0 b0i  

has exactly one eigenvalue inside the unit circle. It can be verified that one eigenvalue equals b0i and the remaining two eigenvalues are given by those of

the characteristic polynomial

p(µ)_≡µ2₋µ(1 +β−1 +λϕβ−1) +β−1.

It is easy to check that p(0)>0 andp(1)<0 so that one eigenvalue of p(µ) is between 0 and 1 and the other one exceeds 1. Note that for a determinate

MSV solution, we must have_|b0

i|<1 so that exactly two eigenvalues ofB−

1 1 Bˆ2

are inside the unit circle. The arguments show that REE is indeterminate with CIRS rule II for all horizons and structural parameters.

We now turn to an analysis of E-stability of the system (38), which is formally the same as in the previous section. One of the necessary conditions for E-stability is that the matrix ˆ_z+ ˆ_z¯b, where

ˆ z+ ˆ_z¯b=   1 ϕ b 0 x+ϕb0π λ β+λϕ λb0 x+ (β+λϕ)b0π 0 0 0  ,

has eigenvalues with real parts less than one. It is easy to see that one eigenvalue of ˆ_z+ ˆ_z¯bis zero and the remaining two are given by those of the matrix A in (6). A has an eigenvalue more than 1 so that all MSV solutions of the model (38) are necessarily E-unstable. This result is again independent of the horizon used by the bank and structural parameters.

C

Details for the Inertial Model

As in Section 2.2, we can write the model with inertia in matrix form as

yt = A1Et∗yt+1+L1yt−1+Bwt+Dit, (39)

wt = V wt−1+vt,

where yt= (xt,πt)0, wt= (gt, ut)0,vt= (˜gt,u˜t)0 and the matrices are

A1 = µ θ ϕ λθ βγ+λϕ ¶ , L1 = µ 1₋θ 0 λ(1₋θ) 1₋γ ¶ . (40)

with B and D as defined before in (6). Strict inflation targeting is defined as before by equations (1) and (7) leading to a form corresponding to (8):

µ y1,t+1 Ety2,t+1 ¶ =Ω1 µ y1,t y2,t ¶ +Ψ1it+ µ vt+1 0 ¶ , (41)

wherey2,t = (xt,πt)0,y1,t= (gt, ut, xlt,πlt)0,vt = (˜gt,u˜t)0,andxlt≡xt−1,πlt≡ πt−1. Also Ω1 =          µ 0 0 0 0 0 0 ρ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 −1θ ϕ θγβ − 1−θ θ ϕ(1−γ) θγβ 1+ϕλγ−1_β−1 θ − ϕ θγβ 0 _−γβ1 0 ₋(1_γβ−γ) _−γβλ _γβ1          ,Ψ1 =         0 0 0 0 ϕ θ 0         .

It is possible to compute the interest rule based on constant interest rate projections in the same way as before. The CIRU K rule I corresponding to

(13) is now it =− Ã K h−1 X j=0 Ωj₁Ψ1 !−1 KΩh₁ µ y1,t y2,t ¶ , (42) with K = (0,0,0,0,0,1).

Proof of Proposition 5: The matrix for checking determinacy, namely,

    −βγ1 1+βγ(γ−1) βγλ 0 γ−1 βγλ − λ βγ 1 βγ 0 γ−1 βγ 1 0 0 0 0 1 0 0    

has all eigenvalues equal to zero. However, since there are two free and two pre-determined variables, determinacy requires exactly two eigenvalues inside the unit circle. We now show that even though indeterminacy prevails, there exists a unique MSV solution when h= 2.

Plugging the interest rule, (35), with the coefficients (36), into the system (39), we get the reduced form system

yt = AfEt∗yt+1+Alyt−1+Awwt, (43) Af = Ã −(1₋γ)−1 1−_λβγ₍₁(1_−γ−₎γ) −λ(1₋γ)−1 ₍₁ −γ)−1 ! , Al= µ 0 ₋λ−1(1₋γ) 0 0 ¶ Aw = µ 0 [λ(γ₋1)]−1₍₁ −γ+ρ) 0 ₋ρ(1₋γ)−1 ¶ .

Note that the lagged output gap and the gt shock do not appear in the

reduced form system (43); the interest rule has offset both these terms. The MSV solution of (43), consequently, takes the form

xt = ax+bxπt−1+cxut, (44)

πt = aπ +bππt−1 +cπut. (45)

It is easy to verify that there exists a unique MSV solution of this form and it involves ax =aπ = 0,and

bx =−λ−1(1−γ), bπ = 0. (46)

We now check E-stability of this unique MSV solution. Assuming agents have a PLM of the form (44)-(45), they compute their forecasts E∗

txt+1 and Et∗πt+1 and these forecasts used in (43) lead to an ALM of the same form.

If agents use t₋1 data to compute their forecasts, the E-stability conditions for such a system are given by Proposition 10.1 in (Evans and Honkapohja 2001). For the constant term, the eigenvalues corresponding to the following characteristic polynomialp(τ) need to have negative real parts for E-stability.

p(τ) =τ2+τ + βγ

γ₋1

However, p(0) < 0, p(_∞) > 0 which implies that there exists a positive eigenvalue.

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ABOUT THE CDMA

The Centre for Dynamic Macroeconomic Analysis was established by a direct grant from the University of St Andrews in 2003. The Centre funds PhD students and a programme of research centred on macroeconomic theory and policy. Specifically, the Centre is interested in the broad area of dynamic macroeconomics but has a particular interest in a number of specific areas such as: characterising the key stylised facts of the business cycle; constructing theoretical models that can match these actual business cycles; using these models to understand the normative and positive aspects of the macroeconomic policymakers' stabilisation problem; the problem of financial constraints and their impact on short and long run economic outcomes. The Centre is also interested in developing numerical tools for analysing quantitative general equilibrium macroeconomic models (such as developing efficient algorithms for handling large sparse matrices). Its affiliated members are Faculty members at St Andrews and elsewhere with interests in the broad area of dynamic macroeconomics. Its international Advisory Board comprises a group of leading macroeconomists and, ex officio, the University's Principal.

Affiliated Members of the School Prof Jagjit S. Chadha (Director) Dr David Cobham

Dr Laurence Lasselle Dr Peter Macmillan Prof Charles Nolan Dr Gary Shea Prof Alan Sutherland Dr Christoph Thoenissen Senior Research Fellow

Prof Andrew Hughes Hallett, Professor of Economics, Vanderbilt University. Research Affiliates

Prof Keith Blackburn, Manchester University. Dr Luisa Corrado, Università degli Studi di Roma. Prof Huw Dixon, York University

Dr Sugata Ghosh, Cardiff University Business School.

Dr Aditya Goenka, Essex University. Dr Campbell Leith, Glasgow University. Dr Richard Mash, New College, Oxford. Prof Patrick Minford, Cardiff Business School. Dr Gulcin Ozkan, York University.

Prof Joe Pearlman, London Metropolitan University.

Prof Neil Rankin, Warwick University. Prof Lucio Sarno, Warwick University. Prof Eric Schaling, Rand Afrikaans University. Dr Frank Smets, European Central Bank.

Dr Robert Sollis, Durham University.

Dr Peter Tinsley, George Washington University and Federal Reserve Board.

Dr Mark Weder, Humboldt Universität zu Berlin. Research Associates Mr Nikola Bokan Mr Vladislav Damjanovic Mr Michal Horvath Ms Elisa Newby Mr Qi Sun Mr Alex Trew Advisory Board

Prof Sumru Altug, Koç University. Prof V V Chari, Minnesota University. Prof Jagjit S. Chadha, St Andrews University. Prof John Driffill, Birkbeck College London. Dr Sean Holly, Director of the Department of

Applied Economics, Cambridge University. Prof Seppo Honkapohja, Cambridge University. Dr Brian Lang, Principal of St Andrews University. Prof Anton Muscatelli, Glasgow University. Prof Charles Nolan, St Andrews University. Prof Peter Sinclair, Birmingham University and

Bank of England.

Prof Stephen J Turnovsky, Washington University. Mr Martin Weale, CBE, Director of the National

Institute of Economic and Social Research. Prof Michael Wickens, York University. Prof Simon Wren-Lewis, Exeter University.

THE CDMAINAUGURAL CONFERENCE 2004

The Inaugural CDMA Conference was held in St. Andrews on the 17th and 18th of September 2004. The list of delegates attending, and the group photo, can be found here.

PAPERS PRESENTED AT THE CONFERENCE, IN ORDER OF PRESENTATION:

Title Author(s) (presenter in bold)

A Critique of rule-of-thumb/indexing Microfoundations for inflation persistence

Richard Mash (Oxford)

Fiscal and Monetary Policy Interactios in a New Keynesian Model with Liquidity Constraints

V. Anton Muscatelli (Glasgow), Patrizio Tirelli (Milano-Bicocca) and Carmine Trecroci (Brescia)

Inflation Persistence as Regime Change

in a Classical Macro Model Patrick Minford (Cardiff and CEPR)Eric Nowell (Liverpool), Prakriti Sofat , (Cardiff) and Naveen Srinivasan (Cardiff)

Habit Formation and Interest Rate

Smoothing Luisa Corrado (Rome ‘Tor Vergata’)Sean Holly (Cambridge) and

A Model of Job and Worker Flow Nobuhiro Kiyotaki (LSE) and Richard Lagos (FRB of Minneapolis and New York)

The Specification of Monetary Policy

Inertia in Empirical Taylor Rules John Driffill (Birkbeck, London)Zeno Rotondi (Ferrera and Capitalia) and

Inequality and Industrialization Parantap Basu (Durham) and Alessandra Guariglia (Nottingham)

Public Expenditures, Bureaucratic

Corruption and Economic Development Keith Blackburn (Manchester)Bose (Wisconsin) and M. Emanrul Haque , Niloy (Nottingham)

On the Consumption-Real Exchange

Rate Anomaly Gianluca Benigno (LSE and CEPR) and Christoph Thoenissen (St Andrews)

The Issue of Persistence in DGE Models

with Heterogeneous Taylor Contracts Huw Dixon (York) and Engin Kara (York

Performance of Inflation Targeting Based

on Constant Interest Rate Projections Seppo Honkapohja (Cambridge)Kaushik Mitra (Royal Holloway, London) and See also the CDMA Working Paper series.